Stokes matrices, character varieties, and points on spheres (with Junho Peter Whang)
Yu-Wei Fan
University of California, Berkeley
Algebra and Algebraic Geometry seminar at UBC February 2021
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 1 / 29 Upshots:
1. Establish connections among braid group actions and natural braid group invariants on the following spaces:
SL2(C)-character varieties of certain Riemann surfaces, space of Stokes matrices, moduli of points on spheres.
2. Obtain certain finiteness results for integral Stokes matrices, via arithmetic dynamics results on character varieties.
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 2 / 29 Stokes matrices and exceptional collections
Stokes matrices V (r) = {unipotent upper triangular r × r matrix}. Exceptional collections of triangulated categories are natural sources of examples of integral Stokes matrices.
Let D be a triangulated category. A collection E = {E1,..., Er } is called exceptional if • Hom (Ei , Ei ) = C[0] for all i, • Hom (Ei , Ej ) = 0 for i > j.
Its Gram matrix (χ(Ei , Ej ))1≤i,j≤r is a Stokes matrix, where
X k k χ(Ei , Ej ) = (−1) dim Hom (Ei , Ej ). k
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 3 / 29 Braid group actions on V (r) Braid group on r-strands:
Br = hσ1, . . . , σr−1|σi σi+1σi = σi+1σi σi+1, σi σj = σj σi for |i − j| ≥ 2i .
Br -actions: On V (r): Ii−1 Ii−1 σi si,i+1 −1 si,i+1 1 s −→ s . 1 0 −1 0 Ir−i−1 Ir−i−1
Categorical level: (left) mutations
σi {E1,..., Er } −→{E1,..., Ei−1, LEi Ei+1, Ei , Ei+2,..., Er }.
• [1] (LE F → Hom (E, F ) ⊗ E → F −→)
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 4 / 29 Braid group invariants on V (r) It’s not hard to check that the conjugacy class of −s−1sT is invariant under the Br -action on V (r).
Let E = {E1,..., Er } be a full exceptional collection of D, and denote sE its Gram matrix. Then the induced automorphism of SerreD on num −1 T K0 (D) is given by sE sE , with respect to the basis {[E1],..., [Er ]}. Problem: Classify full exceptional collections of D up to mutations?
Problem: Classify integral Stokes matrices up to Br -actions in
−1 T Vp(r) = {s : det(λ + s s ) = p(λ)} ⊆ V (r)?
Reults:
r = 3: when disc(p) 6= 0, Vp(3) contains at most finitely many integral B3-orbits (can be proved by Markoff descent argument)
r = 4: when disc(p) 6= 0, Vp(4) contains at most finitely many integral B4-orbits (F.–Whang)
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 5 / 29 Markoff numbers
A Markoff number is a positive integer x, y or z that satisfies the Diophantine equation x2 + y 2 + z2 = xyz.
3 is a Markov number since (x, y, z) = (3, 3, 3) is a solution. Vieta involution:(x, y, z) → (yz − x, y, z). First few Markov numbers: 3, 6, 15, 39, 87,....
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 6 / 29 3 × 3 Stokes matrices and Markoff-type equation
1 x z s = 0 1 y 0 0 1
The characteristic polynomial of −s−1sT is given by
2 pk (λ) = (λ + 1)(λ − kλ + 1),
where k = x2 + y 2 + z2 − xyz − 2.
The braid group B3-action on V (3) gives the Vieta involution,
e.g.: (x, y, z) 7→ (x, z, xz − y).
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 7 / 29 Markoff-type equation and one-holed torus
∼ 3 Vogt–Fricke: Hom(π1(Σ1,1), SL2(C)) // SL2(C) −→ Cx,y,z , with coordinates given by
x = tr(ρ(a)), y = tr(ρ(b)), z = tr(ρ(ab)). The boundary trace is
k := tr(ρ(d)) = x2 + y 2 + z2 − xyz − 2.
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 8 / 29 Vieta involutions and Dehn twists
∗ Dehn twist along a acts by τa :(x, y, z) 7→ (x, z, xz − y), since 2 tr(A B) = tr(A)tr(AB) − tr(B) for any A, B ∈ SL2. Similarly, the Dehn twists along b and ab correspond to the other two Vieta involutions.
B3 = hτa, τb|τaτbτa = τbτaτbi form a braid group, and the boundary trace k is a B3-invariant.
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 9 / 29 We have seen a connection between
one-holed torus ←→ 3 × 3 Stokes matrix
Dehn twists ←→ mutations( B3-actions) −1 T boundary trace ←→ −s s (B3-invariants)
Xk (Σ1,1, SL2(C)) ' Vpk (3) Here Xk (Σ1,1, SL2) = {ρ ∈ X (Σ1,1, SL2): trρ(d) = k} and −1 T Vpk (3) = {s ∈ V (3): det(λ + s s ) = pk (λ)} are B3-invariant subvarieties.
Can we generalize these?
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 10 / 29 Two-holed torus
Let ρ ∈ X = Hom(π1(Σ1,2), SL2(C)), and define
a = trρ(α1), b = trρ(α2), c = trρ(α3), d = trρ(α1α2α3), e = trρ(α1α2), f = trρ(α2α3).
Xk1,k2 (Σ1,2, SL2(C)) with fixed boundary traces k1 := trρ(β1) and 6 k2 := trρ(β2) is a 4-dimensional subvariety of Ca,b,c,d,e,f defined by
ac + bd − ef = k1 + k2 2 2 2 a + b + ··· + f − abe − adf − bcf − cde + abcd − 4 = k1k2
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 11 / 29 4 × 4 Stokes matrices
1 a e d 0 1 b f s = 0 0 1 c 0 0 0 1 The characteristic polynomial of −s−1sT is given by
4 3 2 2 2 pk1,k2 (λ) = λ − k1k2λ + (k1 + k2 − 2)λ − k1k2λ + 1
where
k1 + k2 = ac + bd − ef , 2 2 2 k1k2 = a + b + ··· + f − abe − adf − bcf − cde + abcd − 4.
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 12 / 29 Question: Space Braid invariants SL2(C)-character varieties of surfaces boundary monodromy x x ?? ?? y y space of Stokes matrices −s−1sT
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 13 / 29 The goal of this talk is to explain the following picture:
Space Braid invariants SL2(C)-character varieties of surfaces boundary monodromy k k moduli space of points on complex affine 3-sphere Coxeter invariants , → l space of Stokes matrices −s−1sT
Next, we’ll introduce the moduli space of points on spheres, braid group actions on them, and natural braid group invariants (the Coxeter invariants).
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 14 / 29 Moduli space of points on spheres m S(m) := complex affine hypersurface in C defined by 2 2 hx, xi = x1 + ··· + xm = 1. Define the moduli space of r points on S(m) to be
A(r, m) := S(m)r // SO(m)
where SO(m) acts diagonally.
https://world-processor.com
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 15 / 29 Quandles and braid group actions
Definition A quandle is a set X and a binary operation /: X × X → X such that: x / x = x and x / −: X → X is a bijection for any x ∈ X . x / (y / z) = (x / y) / (x / z) for any x, y, z ∈ X .
r If (X ,/) is a quandle, then Br acts on X by the following moves:
σi (x1,..., xr ) = (x1,..., xi−1, xi / xi+1, xi , xi+2,..., xr ), 1 ≤ i ≤ r − 1.
Example: r = 3.
σ1σ2σ1(x1, x2, x3) = ((x1 / x2) / (x1 / x3), x1 / x2, x1),
σ2σ1σ2(x1, x2, x3) = (x1 / (x2 / x3), x1 / x2, x1).
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 16 / 29 Examples of quandles
Example (Group quandle) Let G be a group. Then u / v := uv −1u gives a quandle structure on G.
Example (Sphere quandle)
For any u ∈ S(m), define su ∈ O(m) by su(v) := 2 hu, vi u − v. Then u / v := su(v) gives a quandle structure on S(m).
It’s easy to check that the orthogonal group O(m) acts on S(m) by r quandle automorphisms. Therefore the Br -action on S(m) naturally descends to a Br -action on the moduli space of points on spheres
A(r, m) = S(m)r // SO(m).
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 17 / 29 Clifford algebras
Let q be a non-degenerate quadratic form over V /C. Then the Clifford algebra is ⊕∞ V ⊗i Cl(V , q) := i=0 hv ⊗ v − q(v): v ∈ V i In the Clifford algebra: For u ∈ S(q), we have u⊗2 = q(u) = 1. −1 For u, v ∈ S(q), we have su(v) = u ⊗ v ⊗ u .
The Pin group Pin(q) is the closed algebraic subgroup of Cl(q)× generated by S(q).
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 18 / 29 Coxeter invariants of A(r, q) Consider the map
r c : S(q) → Pin(q), (u1,..., ur ) 7→ u1 ⊗ · · · ⊗ ur .
It is Br -invariant:
c(σi u) = u1 ⊗ · · · ⊗ sui (ui+1) ⊗ ui ⊗ ui+2 ⊗ · · · ⊗ ur −1 = u1 ⊗ · · · ⊗ (ui ⊗ ui+1 ⊗ ui ) ⊗ ui ⊗ ui+2 ⊗ · · · ⊗ ur = u1 ⊗ · · · ⊗ ur = c(u).
The Coxeter invariant of A(r, q) to be the Br -invariant morphism
c : A(r, q) → Pin(q) // SO(q), [u1,..., ur ] 7→ [u1 ⊗ · · · ⊗ ur ] For each P ∈ Pin(q) // SO(q), denote
−1 AP (r, q) := c (P).
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 19 / 29 Exceptional isomorphisms
Theorem (F.–Whang, 2020)
Write r = 2g + n ≥ 3 where n ∈ {1, 2}. We have a Br -equivariant isomorphism ∼ AP (r, 4) = Xk (Σg,n, SL2(C)), where the Coxeter invariant P determine the boundary monodromy k, and vice versa. Note: The definitions of A(r, m) and their Coxeter invariants have nothing to do with any Riemann surface! Corollary (F.–Whang, 2020)
The Br -action on AP (r, 4) extends to an action of the pure mapping class group of Σg,n.
Similar statements hold for A(r, 1) and A(r, 2). The proof relies on the existence of group structure for spheres of dimension 0, 1, 3.
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 20 / 29 Braid actions and boundary curves of Σg,n
r = 2g + n, where n ∈ {1, 2}.
π1(Σg,n) is a free group of rank 2g + n − 1 = r − 1.
Dehn twists along α1, . . . , αr−1 generate the braid group Br . The boundary curve(s) are homotopic to: −1 I r odd: (α1α3 ··· αr−2)(α1α2 ··· αr−1) (α2α4 ··· αr−1). −1 I r even: α1α3 ··· αr−1 and (α1α2 ··· αr−1) (α2α4 ··· αr−2).
Br acts on X (Σg,n, G) via Dehn twists, and the conjugacy classes of the boundary curve(s) give Br -invariants.
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 21 / 29 Sketch of proof of main theorem
Identifying the spaces: 2 2 2 2 Over C, the quadratic form x1 + x2 + x3 + x4 ∼ x1x4 − x2x3. ∼ −1 S(4) = SL2 carries an action of SL2 × SL2 by (g1, g2) · u = g1ug2 . We have isomorphisms
∼ r ∼ r X (Σg,n.SL2) = SL2 // (SL2 × SL2) = S(4) // SO(4) = A(r, 4)
ρ 7→ [ρ(α1α2 ··· αr−1), ρ(α2 ··· αr−1), . . . , ρ(αr−1), 1]. Braid group equivariance of the isomorphisms:
Dehn twists along α1, . . . , αr−1 are compatible with Br -actions on r SL2 // (SL2 × SL2) induced by group quandle of SL2. −1 ∼ uv u = su(v) for any u, v ∈ S(4) = SL2.
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 22 / 29 Sketch of proof of main theorem (cont’d)
Match boundary monodromy with Coxeter invariants: ∼ 2 2 2 Pin(4) = SL2 t SL2ι with ι = 1 and (g1, g2)ι = ι(g2, g1). In particular, the image of u ∈ S(4) is (u, u−1)ι. ∼ 2 Pin(4) // SO(4) = W (SL2) t W (SL2), where W (SL2) = SL2 // SL2. −1 −1 −1 I r odd: [u1 ⊗ · · · ⊗ ur ] 7→ [u1u2 ··· ur u1 u2 ··· ur ]. −1 −1 −1 I r even: [u1 ⊗ · · · ⊗ ur ] 7→ ([u1u2 ··· ur ], [u1 u2 ··· ur ]). Example: r = 3.
−1 −1 −1 c([ρ(α1α2), ρ(α2), 1]) = [ρ(α1α2)ρ(α2) 1ρ(α1α2) ρ(α2)1 ] −1 −1 = [ρ(α1α2 α1 α2)].
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 23 / 29 Connect with Stokes matrices
Idea: Gram matrix of points on sphere is a symmetric matrix with 1’s on the diagonal ↔ (s + sT )/2. Define V (r, m) ⊆ V (r) to be the subset of Stokes matrices such that rank(s + sT ) ≤ m. By invariant theory for the orthogonal group, there is an isomorphism S(m)r // O(m) =∼ V (r, m) given by: 1 2 hv1, v2i · · · 2 hv1, vr i 0 1 ··· 2 hv2, vr i [(v ,..., v )] 7→ . 1 r ...... . . . . 0 ··· 0 1
It’s not hard to check that this is a Br -equivariant isomorphism.
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 24 / 29 Coxeter invariants on V (r) and −s−1sT
We define the Coxeter invariant on V (r, m) to be the composition
c : V (r, m) −→∼ S(m)r // O(m) → Pin(m) // O(m).
We show that the composition
∼ r V (r) = S(r) // O(r) → Pin(r) // O(r) O(r) // O(r) takes a Stokes matrix s to the conjugacy class of −s−1sT . Hence we can use the Coxeter invariant of points on spheres to give a refinement of the conjugacy class of −s−1sT of Stokes matrix s.
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 25 / 29 Character varieties and Stokes matrices
We have a sequence of Br -equivariant morphisms
∼ r 2:1 r ∼ X (Σg,n, SL2(C)) = S(4) // SO(4) −−→ S(4) // O(4) = V (r, 4) ,→ V (r).
Moreover, suppose s ∈ V (r) is the image of ρ ∈ X (Σg,n, SL2(C)). Then the boundary trace(s) of ρ and the characteristic polynomial p(λ) of −s−1sT are related as follows. When r is odd: Let k be the boundary trace of ρ, then
p(λ) = (λ2 − kλ + 1)(λ + 1)(λ − 1)r−3.
When r is even: Let k1, k2 be the boundary traces of ρ, then
4 3 2 2 2 r−4 p(λ) = λ − k1k2λ + (k1 + k2 − 2)λ − k1k2λ + 1 (λ − 1) .
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 26 / 29 Diophantine result on Xk (Σg,n, SL2)(Z)
A point ρ ∈ Xk (Σg,n, SL2) is integral if its monodromy trace along every essential simple closed curve is integral.
Whang, 2017: The non-degenerate integral points of Xk (Σg,n, SL2) consist of finitely many mapping class group orbits.
The morphism Xk (Σg,n, SL2) → Vp(r) takes an integral point to an integral Stokes matrix: 1 trρ(α1) trρ(α1α2) ··· trρ(α1 ··· αr−1) 1 tr(α2) ··· trρ(α2 ··· αr−1) 1 ··· trρ(α3 ··· αr−1) ρ 7→ . .. . . . 1
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 27 / 29 Application to Stokes matrices and exceptional collections Corollary (F.–Whang, 2020)
Let p be a degree 4 polynomial with disc(p) 6= 0. Then Vp(4) contains at most finitely many integral B4-orbits.
Sketch of proof: By the main theorem, there is a decomposition ∼ G ∼ G Vp(4) = AP (4, 4) = Xk (Σ1,2, SL2), P∈Pin(4)//SO(4) k
and the B4-action coincides with the Γ(Σ1,2)-action. The corollary follows from analysis of degenerate integral points on Xk (Σ1,2, SL2). Corollary (F.–Whang, 2020) Let D be a triangulated category admitting a full exceptional collection of num legnth 4. Assume that disc(det(λ + SerreD )) 6= 0. Then there is a finite list of Stokes matrices of rank four such that, up to mutations, the Gram matrix of any full exceptional collection of D belongs to this list.
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 28 / 29 Thank you for your attention!
Reference: Fan–Whang. Surfaces, braids, Stokes matrices, and points on spheres. arXiv:2009.07104.
Yu-Wei Fan Stokes matrices, surfaces, spheres February 2021, UBC 29 / 29